Problem: The following function gives the cost, in dollars, of producing $x$ gallons of wood stain: $C(x)=0.0004x^3-0.001x^2+0.1x+3200$ What is the instantaneous rate of change of the cost when $100$ gallons are produced? Choose 1 answer: Choose 1 answer: (Choice A) A $11.9$ dollars (Choice B) B $11.9$ dollars per gallon (Choice C) C $3600$ dollars (Choice D) D $3600$ dollars per gallon
Explanation: Understanding the problem The function that represents the instantaneous rate of change of $C(x)$ is its derivative, $C'(x)$. Therefore, the instantaneous rate of change of the cost when $100$ gallons are produced is $C'(100)$. Let's find $C'(x)$ and evaluate it at $x=100$. Finding $C'(x)$ $C'(x)=0.0012x^2-0.002x+0.1$ Finding $C'(100)$ $\begin{aligned} C'(100)&=0.0012(100)^2-0.002(100)+0.1 \\\\ &=11.9 \end{aligned}$ Interpreting units $C(x)$ is the cost in ${\text{dollars}}$ for $x$ ${\text{gallons}}$. Therefore, we measure its rate of change in ${\text{dollars}}$ per ${\text{gallon}}$. In conclusion, the instantaneous rate of change of the cost when $100$ gallons are produced is $11.9$ dollars per gallon. The rate of change is positive because the cost is increasing.